We study the following nonlinear eigenvalue problem with nonlinear Robin boundary condition
\left\{ {\matrix{ { - {\Delta _p}u = \lambda {{\left| u \right|}^{p - 2}}u\,\,\,in\,\,\Omega ,} \hfill \cr {{{\left| {\nabla u} \right|}^{p - 2}}\nabla u.v + {{\left| u \right|}^{p - 2}}u = 0\,\,\,on\,\,\Gamma .} \hfill \cr } } \right.
We successfully investigate the existence at least of one nondecreasing sequence of positive eigenvalues λn↗∞. To this end we endow W1,p(Ω) with a norm invoking the trace and use the duality mapping on W1,p (Ω) to apply mini-max arguments on C1-manifold.